Abstract:

Classical spectral methods are subject to two fundamental limitations:  they only can account for covariance-related serial dependencies, and they require second-order stationarity.  Much attention has been devoted lately to {\em quantile} (copula-based) {\em spectral methods} that go beyond traditional covariance-based serial dependence features. At the same time, covariance-based methods relaxing stationarity  into  much weaker {\it local stationarity} conditions have been developed for a variety of time-series models. Here, we are combining those two approaches by proposing copula-based spectral methods for locally stationary processes. We therefore introduce a time-varying version of the copula spectra that have been  recently proposed in the literature, along with a suitable local lag-window estimator. We propose a new definition of local {\it strict} stationarity that allows  us to handle completely general non-linear processes without any moment assumptions, thus accommodating our copula-based concepts and methods. We establish a central limit theorem for the new estimators, and illustrate the power of the proposed methodology by means of a simulation study. Moreover,  real-data applications demonstrate that the new approach detects   important variations in  serial dependence structures both across time and across quantiles. Such variations remain completely undetected, and are actually undetectable, via classical covariance-based spectral methods.

Based on joint work with Stefan Birr (Ruhr-Universit\” at Bochum), Holger Dette (Ruhr-Universit\” at Bochum), Tobias Kley (London School of Economics) and Stanislav Volgushev (University of Toronto).